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Global rigidity of 3-dimensional cone-manifolds

Hartmut Weiss
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Abstract

We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles $\leq π$ (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles $\leq π$, possibly with boundary consisting of totally geodesic hyperbolic turnovers. To that end we first generalize our local rigidity result to the setting of hyperbolic cone-3-manifolds of finite volume as above. We then use the geometric techniques developed by Boileau, Leeb and Porti to deform the cone-manifold structure to a complete non-singular or a geometric orbifold structure, where global rigidity holds due to Mostow-Prasad rigidity in the hyperbolic case, resp. a result of de Rham in the spherical case. This strategy has already been implemented successfully by Kojima in the compact hyperbolic case if the singular locus is a link using Hodgson-Kerckhoff local rigidity.

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Hartmut Weiss. 2007-02-14. Global rigidity of 3-dimensional cone-manifolds. https://arxiv.org/abs/math/0504117

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